In →Inormal-→absentI\to\!I, when the formula BBB is deduced from the assumption or hypothesisAAA, we conclude with the formula A→Bnormal-→ABA\to B. Once this conclusion is reached, AAA is superfluous and therefore removed, as it is embodied in the formula A→Bnormal-→ABA\to B. This is often encountered in mathematical proofs: if we want to prove A→Bnormal-→ABA\to B, we first assume AAA, then we proceed with the proof and reach BBB, and therefore A→Bnormal-→ABA\to B. Simiarly, in ∨EE\lor\!E, if CCC can be concluded from AAA and from BBB individually, then CCC can be concluded from anyone of them, or A∨BABA\lor B, without the assumptions AAA and BBB individually.

Intuitionistic propositional logic as defined by the natural deduction system above is termed NJ. Derivations and theorems for NJ are defined in the usual manner like all natural deduction systems, which can be found here. Some of the theorems of NJ are listed below:

For example, ⊢A→¬⁢¬⁢Aprovesabsentnormal-→AA\vdash A\to\neg\neg A as
\prooftree\AxiomC[A]2subscriptA2[A]_{2}\AxiomC[¬⁢A]1subscriptA1[\neg A]_{1}\RightLabel(→E)normal-→absentE(\to\!E)\BinaryInfC⟂perpendicular-to\perp\RightLabel(→I)1subscriptfragmentsnormal-(normal-→Inormal-)1(\to\!I)_{1}\UnaryInfC¬A→⟂fragmentsAnormal-→perpendicular-to\neg A\to\perp\RightLabel(→I)2subscriptfragmentsnormal-(normal-→Inormal-)2(\to\!I)_{2}\UnaryInfCA→(¬A→⟂)fragmentsAnormal-→fragmentsnormal-(Anormal-→perpendicular-tonormal-)A\to(\neg A\to\perp)
and A→(¬A→⟂)fragmentsAnormal-→fragmentsnormal-(Anormal-→perpendicular-tonormal-)A\to(\neg A\to\perp) is just A→¬⁢¬⁢Anormal-→AAA\to\neg\neg A. Also, ⊢¬⁢¬⁢¬⁢A→¬⁢Aprovesabsentnormal-→AA\vdash\neg\neg\neg A\to\neg A, as
\prooftree\AxiomC[A]1subscriptA1[A]_{1}\AxiomC

A→¬¬A

\RightLabel

(→E)normal-→absentE(\to\!E)\BinaryInfC¬⁢¬⁢AA\neg\neg A\AxiomC[¬⁢¬⁢¬⁢A]2subscriptA2[\neg\neg\neg A]_{2}\RightLabel(→E)normal-→absentE(\to\!E)\BinaryInfC⟂perpendicular-to\perp\RightLabel(→I)1subscriptfragmentsnormal-(normal-→Inormal-)1(\to\!I)_{1}\UnaryInfC¬⁢AA\neg A\RightLabel(→I)2subscriptfragmentsnormal-(normal-→Inormal-)2(\to\!I)_{2}\UnaryInfC¬⁢¬⁢¬⁢A→¬⁢Anormal-→AA\neg\neg\neg A\to\neg A
The subscripts indicate that the discharging of the assumptions at the top correspond to the applications of the inference rules→Inormal-→absentI\to\!I at the bottom. The box around A→¬⁢¬⁢Anormal-→AAA\to\neg\neg A indicates that the derivation of A→¬⁢¬⁢Anormal-→AAA\to\neg\neg A has been embedded (as a subtree) into the derivation of ¬⁢¬⁢¬⁢A→¬⁢Anormal-→AA\neg\neg\neg A\to\neg A.

Remark. If ¬\neg were introduced as a primitive logical symbol instead of it being as “defined”, then we need to have inference rules for ¬\neg as well, one of which is introduction ¬⁢II\neg I, and the other elimination ¬⁢EE\neg E: